Question

The binary operation $\ast$ defined on $R$ as $a\ast b=1\forall a,b\in R$ is

• A. both associative and commutative
• B. commutative but not associative
• C. neither associative nor commutative
• D. associative but not commutative

Answer 1

The correct option is A both associative and commutative

$a\ast b=1\forall a,b\in R$
$\text{Clearly, by definition,}a\ast b=b\ast a\forall a,b\in R$
Hence $\ast$ is commutative.

$\text{Let}a,b,c\in R$
$(a\ast b)\ast c=1\ast c=1$
$a\ast (b\ast c)=a\ast 1=1$
Hence $\ast$ is associative.